Acceleration: Difference between revisions

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The following formula describes the velocity of a ship accelerating from a standstill after a particular time:

<math>V_{\rm{t}} = V_{\rm{max}} \times \left( 1 - e^{ \Large\frac{-t\times10^6}{I\times M} } \right)</math>

:<math>V_{\rm{t}} = V_{\rm{max}} \times \left( 1 - e^{ \Large\frac{-t\times10^6}{I\times M} } \right)</math>

where:

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Rearranging the formula for t we arrive at the formula for time taken to accelerate from zero to V:

<math>\displaystyle t_~~{\rm~~{V}} = -I \times M \times 10^{-6} \times \ln\left( 1 - \frac{V}{V_{\rm{max}} } \right)</math>

:<math>\displaystyle t_{V} = -I \times M \times 10^{-6} \times \ln\left( 1 - \frac{V}{V_{\rm{max}} } \right)</math>

Where ''tV'' is the time to accelerate to velocity ''V'' in seconds. Note that at ''V'' = ''Vmax'', 1 - ''V'' / ''Vmax'' = 0, but ln 0 is undefined, so in theory it takes infinite time to reach maximum speed (technically, the limit of ''tV'' as ''V'' approaches ''Vmax'' is positive infinity). In practice the game simulation is not perfectly accurate and it actually takes finite time to reach maximum speed to within whatever precision the simulation uses. (Strictly speaking, velocity is a vector, so it has both direction and magnitude, but the game is really only concerned with its absolute value (i.e. the magnitude part) more commonly called 'speed'.)

@@ Line 28: / Line 28: @@
 The following formula describes the velocity of a ship accelerating from a standstill after a particular time:
-<math>V_{\rm{t}} = V_{\rm{max}} \times \left( 1 - e^{ \Large\frac{-t\times10^6}{I\times M} }  \right)</math>
+:<math>V_{\rm{t}} = V_{\rm{max}} \times \left( 1 - e^{ \Large\frac{-t\times10^6}{I\times M} }  \right)</math>
 where:
@@ Line 57: / Line 57: @@
 Rearranging the formula for t we arrive at the formula for time taken to accelerate from zero to V:
-<math>\displaystyle t_{\rm{V}} = -I \times M \times 10^{-6} \times \ln\left( 1 - \frac{V}{V_{\rm{max}} } \right)</math>
+:<math>\displaystyle t_{V} = -I \times M \times 10^{-6} \times \ln\left( 1 - \frac{V}{V_{\rm{max}} } \right)</math>
 Where ''t<sub>V</sub>'' is the time to accelerate to velocity ''V'' in seconds. Note that at ''V'' = ''V<sub>max</sub>'', 1 - ''V'' / ''V<sub>max</sub>'' = 0, but ln 0 is undefined, so in theory it takes infinite time to reach maximum speed (technically, the limit of ''t<sub>V</sub>'' as ''V'' approaches ''V<sub>max</sub>'' is positive infinity). In practice the game simulation is not perfectly accurate and it actually takes finite time to reach maximum speed to within whatever precision the simulation uses. (Strictly speaking, velocity is a vector, so it has both direction and magnitude, but the game is really only concerned with its absolute value (i.e. the magnitude part) more commonly called 'speed'.)